Frequency of finger looking during finger counting is related to children's working memory capacities

Finger counting can be useful in solving arithmetic problems, noticeably because it reduces the working memory demand of mental calculations. However, proprioceptive information might not be sufficient to keep track of the number of fingers raised during problem solving, and visual input may play an important role in this process. The present study was designed to address this question and shows that 8-year-old children look at their fingers in 60% of the trials during finger counting when solving additive problems. Moreover, our results reveal that the frequency of finger looking is negatively correlated with working memory capacities and is higher for more difficult problems. These findings suggest that finger looking is recruited in managing the cognitive demand of the arithmetic task, probably by providing additional external cues to monitor the number of steps that have to be incremented during finger counting.     

The impact of finger counting habits on arithmetic in adults and children

Here, we explored the impact of finger counting habits on arithmetic in both adults and children. Two groups of participants were examined, those that begin counting with their left hand (left-starters) and those that begin counting with their right hand (right-starters). For the adults, performance on an addition task in which participants added 2 two-digit numbers was compared. The results revealed that left-starters were slower than right-starters when adding and they had lower forward and backward digit-span scores. The children (aged 5–12) showed similar results on a single-digit timed addition task—right-starters outperformed left-starters. However, the children did not reveal differences in working memory or verbal and non-verbal intelligence as a function of finger counting habit. We argue that the motor act of finger counting influences how number is represented and suggest that left-starters may have a more bilateral representation that accounts for the slower processing.     

数字记忆广度对策略应用模式的影响──中美儿童计算能力差异的比较研究

本研究旨在了解中美儿童在策略运用模式上的差异是否与数字记忆广度有关。为此,分别测验了中美幼儿园儿童的加法技能,数字记忆广度,并通过一组计算机呈现的加法作业评估儿童使用的解题策略。结果表明,中国儿童的基本算术技能和数学记忆广度均超过同龄的美国儿童。在解题时,中国儿童多使用效率较高的言语计算策略,美国儿童则多采用计数手指策略,在言语计数策略和检索策略的加工速度上中国儿童也超过美国儿童。相关分析表明．策略应用模式的文化差异与数字记忆广度有关。比较大的工作记忆容量增加了数字线索可利用的资源,为儿童早期言语计数能力的发展提供了有利条件。     

The development of strategy use in elementary school children: working memory and individual differences

The current study tested the development of working memory involvement in children's arithmetic strategy selection and strategy efficiency. To this end, an experiment in which the dual-task method and the choice/no-choice method were combined was administered to 10- to 12-year-olds. Working memory was needed in retrieval, transformation, and counting strategies, but the ratio between available working memory resources and arithmetic task demands changed across development. More frequent retrieval use, more efficient memory retrieval, and more efficient counting processes reduced the working memory requirements. Strategy efficiency and strategy selection were also modified by individual differences such as processing speed, arithmetic skill, gender, and math anxiety. Short-term memory capacity, in contrast, was not related to children's strategy selection or strategy efficiency.     

Working memory and children's use of retrieval to solve addition problems

This study tested the hypothesis that children with high working memory capacities solve single-digit additions by direct retrieval of the answers from long-term memory more often than do children with low working memory capacities. Counting and reading letter span tasks were administered to groups of third-grade (mean age=107 months) and fourth-grade (mean age=118 months) children who were also asked to solve 40 single-digit additions. High working memory capacity was associated with more frequent use of retrieval and faster responses in solving additions. The effect of span on the use of retrieval increased with the size of the minimum addend. The relation between working memory measures and use and speed of retrieval did not depend on the numerical or verbal nature of the working memory task. Implications for developmental theories of cognitive arithmetic and theories of working memory are discussed.     

The roles of the central executive and visuospatial storage in mental arithmetic: A comparison across strategies

Previous research has demonstrated that working memory plays an important role in arithmetic. Different arithmetical strategies rely on working memory to different extents—for example, verbal working memory has been found to be more important for procedural strategies, such as counting and decomposition, than for retrieval strategies. Surprisingly, given the close connection between spatial and mathematical skills, the role of visuospatial working memory has received less attention and is poorly understood. This study used a dual-task methodology to investigate the impact of a dynamic spatial n-back task (Experiment 1) and tasks loading the visuospatial sketchpad and central executive (Experiment 2) on adults' use of counting, decomposition, and direct retrieval strategies for addition. While Experiment 1 suggested that visuospatial working memory plays an important role in arithmetic, especially when counting, the results of Experiment 2 suggested this was primarily due to the domain-general executive demands of the n-back task. Taken together, these results suggest that maintaining visuospatial information in mind is required when adults solve addition arithmetic problems by any strategy but the role of domain-general executive resources is much greater than that of the visuospatial sketchpad.     

Working memory failures in children with arithmetical difficulties.

A large body of literature has examined the relationship between working memory and arithmetic achievement, but results are still ambiguous. To examine this relationship, we compared the performance of third and fifth graders with arithmetic difficulties (AD) and controls of the same age, grade, and verbal intelligence on a battery of working memory tasks, differentiating between different aspects of working memory. Children with AD scored significantly lower on active working memory tasks requiring manipulation of the to-be-recalled information (Listening Completion task, Corsi Span Backwards, Digit Backwards), but not in passive working memory tasks, requiring the recall of information in the same format in which it had been presented (Digit, Word, and Corsi Forwards Span tasks), nor in tasks involving word processing (word articulation rate, forwards and backwards word spans). A regression analysis showed that the best predictors of differences between AD children and the control group were the Corsi Span Backwards, the Listening Completion task, and the rate of articulation of pseudowords. The analysis of strategies used by children in mental calculation revealed the greater tendency of children with AD to rely on more primitive strategies: finger use never appeared as the most frequent strategy in skilled children, whereas it was the most used strategy in children with AD. Verbal and visual strategies appeared associated with successful performance in third graders, but in fifth grade, the most successful strategy was verbalization.     

Working Memory Failures in Children with Arithmetical Difficulties

A large body of literature has examined the relationship between working memory and arithmetic achievement, but results are still ambiguous. To examine this relationship, we compared the performance of third and fifth graders with arithmetic difficulties (AD) and controls of the same age, grade, and verbal intelligence on a battery of working memory tasks, differentiating between different aspects of working memory. Children with AD scored significantly lower on active working memory tasks requiring manipulation of the to-be-recalled information (Listening Completion task, Corsi Span Backwards, Digit Backwards), but not in passive working memory tasks, requiring the recall of information in the same format in which it had been presented (Digit, Word, and Corsi Forwards Span tasks), nor in tasks involving word processing (word articulation rate, forwards and backwards word spans). A regression analysis showed that the best predictors of differences between AD children and the control group were the Corsi Span Backwards, the Listening Completion task, and the rate of articulation of pseudowords. The analysis of strategies used by children in mental calculation revealed the greater tendency of children with AD to rely on more primitive strategies: finger use never appeared as the most frequent strategy in skilled children, whereas it was the most used strategy in children with AD. Verbal and visual strategies appeared associated with successful performance in third graders, but in fifth grade, the most successful strategy was verbalization.     

Anatomically ordered tapping interferes more with one-digit addition than two-digit addition: a dual-task fMRI study

Fingers are used as canonical representations for numbers across cultures. In previous imaging studies, it was shown that arithmetic processing activates neural resources that are known to participate in finger movements. Additionally, in one dual-task study, it was shown that anatomically ordered finger tapping disrupts addition and subtraction more than multiplication, possibly due to a long-lasting effect of early finger counting experiences on the neural correlates and organization of addition and subtraction processes. How arithmetic task difficulty and tapping complexity affect the concurrent performance is still unclear. If early finger counting experiences have bearing on the neural correlates of arithmetic in adults, then one would expect anatomically and non-anatomically ordered tapping to have different interference effects, given that finger counting is usually anatomically ordered. To unravel these issues, we studied how (1) arithmetic task difficulty and (2) the complexity of the finger tapping sequence (anatomical vs. non-anatomical ordering) affect concurrent performance and use of key neural circuits using a mixed block/event-related dual-task fMRI design with adult participants. The results suggest that complexity of the tapping sequence modulates interference on addition, and that one-digit addition (fact retrieval), compared to two-digit addition (calculation), is more affected from anatomically ordered tapping. The region-of-interest analysis showed higher left angular gyrus BOLD response for one-digit compared to two-digit addition, and in no-tapping conditions than dual tapping conditions. The results support a specific association between addition fact retrieval and anatomically ordered finger movements in adults, possibly due to finger counting strategies that deploy anatomically ordered finger movements early in the development.     

Mental addition in third,fourth, and sixth graders

Research on mental arithmetic has suggested that young children use a counting algorithm for simple mental addition, but that adults use memory retrieval from an organized representation of addition facts. To determine the age at which performance shifts from counting to retrieval, children in grades 3, 4, and 6 were tested in a true/false verification task. Reaction time patterns suggested that third grade is a transitional age with respect to memory structure for addition—half of these children seemed to be counting and half retrieving from memory. Results from fourth and sixth graders implicated retrieval quite strongly, as their results resembled adult RTs very closely. Fourth graders' processing, however, was easily disrupted when false problems were presented. The third graders' difficulties are not due to an inability to form mental representations of number; all three grades demonstrated a strong split effect, indicative of a simpler mental representation of numerical information than is necessary for addition. The results were discussed in the context of memory retrieval versus counting models of mental arithmetic, and the increase across age in automaticity of retrieval processes.     

Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability

Groups of first-grade (mean age = 82 months), third-grade (mean age = 107 months), and fifth-grade (mean age = 131 months) children with a learning disability in mathematics (MD, n = 58) and their normally achieving peers (n = 91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4+3) and complex (16+8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.     

The role of fingers in the development of counting and arithmetic skills

Interactions between fingers and numbers have been reported in the existing literature on numerical cognition. The aim of the present research was to test whether hand interference movements might have an impact on children performance in counting and basic arithmetic problem solving. In Experiment 1, 5-year-old children had to perform both a one-target and a two-target counting task in three different conditions: with no constraints, while making interfering hand movements or while making interfering foot movements. In Experiment 2, first and fourth graders were required to perform addition problems under the same control and sensori-motor interfering conditions. In both tasks, the hand movements caused more disruption than the foot movements, suggesting that finger-counting plays a functional role in the development of counting and arithmetic.     

Visuospatial and verbal memory in mental arithmetic

Working memory allows complex information to be remembered and manipulated over short periods of time. Correlations between working memory and mathematics achievement have been shown across the lifespan. However, only a few studies have examined the potentially distinct contributions of domain-specific visuospatial and verbal working memory resources in mental arithmetic computation. Here we aimed to fill this gap in a series of six experiments pairing addition and subtraction tasks with verbal and visuospatial working memory and interference tasks. In general, we found higher levels of interference between mental arithmetic and visuospatial working memory tasks than between mental arithmetic and verbal working memory tasks. Additionally, we found that interference that matched the working memory domain of the task (e.g., verbal task with verbal interference) lowered working memory performance more than mismatched interference (verbal task with visuospatial interference). Findings suggest that mental arithmetic relies on domain-specific working memory resources.     

Independent contributions of the central executive, intelligence, and in-class attentive behavior to developmental change in the strategies used to solve addition problems

Children's (N=275) use of retrieval, decomposition (e.g., 7=4+3 and thus 6+7=6+4+3), and counting to solve additional problems was longitudinally assessed from first grade to fourth grade, and intelligence, working memory, and in-class attentive behavior was assessed in one or several grades. The goal was to assess the relation between capacity of the central executive component of working memory, controlling for intelligence and in-class attentive behavior, and grade-related changes in children's use of these strategies. The predictor on intercept effects from multilevel models revealed that children with higher central executive capacity correctly retrieved more facts and used the most sophisticated counting procedure more frequently and accurately than their lower capacity peers at the beginning of first grade, but the predictor on slope effects indicated that this advantage disappeared (retrieval) or declined in importance (counting) from first grade to fourth grade. The predictor on slope effects also revealed that from first grade to fourth grade, children with higher capacity adopted the decomposition strategy more quickly than other children. The results remained robust with controls for children's sex, race, school site, speed of encoding Arabic numerals and articulating number words, and mathematics achievement in kindergarten. The results also revealed that intelligence and in-class attentive behavior independently contributed to children's strategy development.     

Switch costs and the operand-recognition paradigm

Experimental research in cognitive arithmetic frequently relies on participants’ self-reports to discriminate solutions based on direct memory retrieval from use of procedural strategies. Given concerns about the validity and reliability of strategy reports, Thevenot et al. in Mem Cogn 35:1344–1352, (2007) developed the operand-recognition paradigm as an objective measure of arithmetic strategies. Participants performed addition or number comparison on two sequentially presented operands followed by a speeded operand-recognition task. Recognition times increased with problem size following addition but not comparison. Thevenot et al. argued that the complexity of addition strategies increases with problem size. A corresponding increase in operand-recognition time occurs because, as problem size increases, working memory contains more numerical distracters. However, because addition is substantially more difficult than comparison, and difficulty increases with problem size for addition but not comparison, their findings could be due to difficulty-related task-switching costs. We repeated Thevenot et al. (Experiment 1) but added a control condition wherein participants performed a parity (odd or even) task instead of operand recognition. We replicated their findings for operand recognition but found robust, albeit smaller, effects of addition problem size on parity judgements. The results indicate that effects of strategy complexity in the operand-recognition paradigm are confounded with task-switching effects, which complicates its application as a precise measure of strategy complexity in arithmetic.     

Patterns of problem‐solving in children's literacy and arithmetic

Patterns of problem‐solving among 5‐to‐7 year‐olds' were examined on a range of literacy (reading and spelling) and arithmetic‐based (addition and subtraction) problem‐solving tasks using verbal self‐reports to monitor strategy choice. The results showed higher levels of variability in the children's strategy choice across Years 1 and 2 on the arithmetic (addition and subtraction) than literacy‐based tasks (reading and spelling). However, across all four tasks, the children showed a tendency to move from less sophisticated procedural‐based strategies, which included phonological strategies for reading and spelling and counting‐all and finger modelling for addition and subtraction, to more efficient retrieval methods from Years 1 to 2. Distinct patterns in children's problem‐solving skill were identified on the literacy and arithmetic tasks using two separate cluster analyses. There was a strong association between these two profiles showing that those children with more advanced problem‐solving skills on the arithmetic tasks also showed more advanced profiles on the literacy tasks. The results highlight how different‐aged children show flexibility in their use of problem‐solving strategies across literacy and arithmetical contexts and reinforce the importance of studying variations in children's problem‐solving skill across different educational contexts.     

Age-related changes in multi-finger interactions in adults during maximum voluntary finger force production tasks

This study aimed to continue our characterization of finger strength and multi-finger interactions across the lifespan to include those in their 60s and older. Building on our previous study of children, we examined young and elderly adults during isometric finger flexion and extension tasks. Sixteen young and 16 elderly, gender-matched participants produced maximum force using either a single finger or all four fingers in flexion and extension. The maximum voluntary finger force (MVF), the percentage contributions of individual finger forces to the sum of individual finger forces during four-finger MVF task (force sharing), and the non-task finger forces during a task finger MVF task (force enslaving), were computed as dependent variables. Force enslaving during finger extension was greater than during flexion in both young and elderly groups. The flexion-extension difference was greater in the elderly than the young adult group. The greater independency in flexion may result from more frequent use of finger flexion in everyday manipulation tasks. The non-task fingers closer to a task finger produced greater enslaving force than non-task fingers farther from the task finger. The force sharing pattern was not different between age groups. Our findings suggest that finger strength decreases over the aging process, finger independency for flexion increases throughout development, and force sharing pattern remains constant across the lifespan.     

Strategy use in mental subtraction determines central executive load

A dual task method was used to examine the relationship between strategy use and working memory load during subtraction problem solving. Undergraduates mentally solved subtraction problems alone and while performing secondary tasks that involved the central executive of working memory. Analyses revealed that a central executive task involving response selection and input monitoring (CRT-R task) interfered more with subtraction problem solving than a task that involved only input monitoring (SRT-R task). Additional analyses showed that the CRT-R task interfered more when participants used a nonretrieval (counting) strategy than a retrieval strategy. These findings suggest that the response selection subcomponent of the central executive is involved during both retrieval-based and non-retrieval-based simple subtraction problem solving but is involved more during the latter.     

工作记忆成分的年龄相关差异对算术策略运用的预测效应

采用选择/无选范式,借助工作记忆成套测验,在两位数乘法估算问题中探讨了工作记忆系统各成分对不同年龄段个体算术策略运用的预测效应。结果显示:(1)工作记忆的不同成分与年龄之间存在明显的相关。表现为,除视空模板成分外,其他各成分得分随着年龄增长而呈现上升趋势;(2)估算策略运用中,年龄与策略选择显著相关,表现为随着年龄增长,策略选择表现明显提高;(3)估算策略运用中,不同年龄个体的工作记忆不同成分和策略选择表现出不同的联系,中央执行均显示出显著的预测效应,语音环路和视空模板的预测效应均不显著。不同年龄个体的工作记忆不同成分对策略执行的预测效应均不显著。上述发现对于深刻理解工作记忆系统在算术认知策略运用中的作用机制具有重要理论含义。